A Morse function f : M → ℝ on a manifold M has only non-degenerate critical points — where df = 0 and the Hessian is non-singular.
As the level set f⁻¹(c) sweeps upward, topology changes only at critical values. Each critical point adds a cell of index equal to the number of negative Hessian eigenvalues.
Morse inequality: β_k ≤ C_k
where β_k = k-th Betti number, C_k = # critical points of index k.
The Euler–Poincaré formula: χ(M) = Σ(−1)^k C_k holds for any Morse function — topology is encoded in critical points.
For a torus: χ = 0 (1 min, 2 saddles, 1 max). Sphere: χ = 2 (1 min, 1 max). Genus-g surface: χ = 2 − 2g.